The aim of this paper is to find a concrete bound for the error involved when approximating the nth Hermite function (in the oscillating range) by an asymptotic formula due to D. Dominici. This bound is then used to study the accuracy of certain approximations to Hermite expansions and to Fourier transforms. A way of estimating an unknown probability density is proposed.

We present a new valuation formula for a generic, multi-period binary option in a multi-asset Black–Scholes economy. The payoff of this so-called M-binary is the most general possible, subject to the condition that a simple analytic expression exists for the present value. Portfolios of M-binaries can be used to statically replicate many European exotics for which there exist closed-form Black–Scholes prices.

The successive over-relaxation (SOR) iteration method for solving linear systems of equations depends upon a relaxation parameter. A well-known theory for determining this parameter was given by Young for consistently ordered matrices. In this paper, for the three-dimensional Laplacian, we introduce several compact difference schemes and analyse the block-SOR method for the resulting linear systems. Their optimum relaxation parameters are given for the first time. Analysis shows that the value of the optimum relaxation parameter of block-SOR iteration is very sensitive for compact stencils when solving the three-dimensional Laplacian. This paper provides a theoretical solution for determining the optimum relaxation parameter in real applications.

We study the H –1-norm of the function 1 on tubular neighbourhoods of curves in ${\mathbb R}^{2}$ . We take the limit of small thickness ε, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit ε → 0, containing contributions from the length of the curve (at order ε 3), the ends (ε 4), and the curvature (ε 5). The second result is a Γ-convergence result, in which the central curve may vary along the sequence ε → 0. We prove that a rescaled version of the H –1-norm, which focuses on the ε 5 curvature term, Γ-converges to the L 2-norm of curvature. In addition, sequences along which the rescaled norm is bounded are compact in the W 1,2-topology. Our main tools are the maximum principle for elliptic equations and the use of appropriate trial functions in the variational characterisation of the H –1-norm. For the Γ-convergence result we use the theory of systems of curves without transverse crossings to handle potential intersections in the limit.

In this paper, we extend the results in the literature for boundary layer flow over a horizontal plate, by considering the buoyancy force term in the momentum equation. Using a similarity transformation, we transform the partial differential equations of the problem into coupled nonlinear ordinary differential equations. We first analyse several special cases dealing with the properties of the exact and approximate solutions. Then, for the general problem, we construct series solutions for arbitrary values of the physical parameters. Furthermore, we obtain numerical solutions for several sets of values of the parameters. The numerical results thus obtained are presented through graphs and tables and the effects of the physical parameters on the flow and heat transfer characteristics are discussed. The results obtained reveal many interesting behaviours that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.

In this paper, we deal with a class of reflected backward stochastic differential equations (RBSDEs) corresponding to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness of the solution for RBSDEs by means of the penalization method. As an application, we give a probabilistic interpretation for the solutions of a class of partial differential-integral inclusions.

Implicit Runge–Kutta methods have a special role in the numerical solution of stiff problems, such as those found by applying the method of lines to the partial differential equations arising in physical modelling. Of particular interest in this paper are the high-order methods based on Gaussian quadrature and the efficiently implementable singly implicit methods.

This is a review of progress made since [R. McKibbin, “An attempt at modelling hydrothermal eruptions”, Proc. 11th New Zealand Geothermal Workshop 1989 (University of Auckland, 1989), 267–273] began development of a mathematical model for progressive hydrothermal eruptions (as distinct from “blasts”). Early work concentrated on modelling the underground process, while lately some attempts have been made to model the eruption jet and the flight and deposit of ejected material. Conceptually, the model is that of a boiling and expanding two-phase fluid rising through porous rock near the ground surface, with a vertical high-speed jet, dominated volumetrically by the gas phase, ejecting rock particles that are then deposited on the ground near the eruption site. Field observations of eruptions in progress and experimental results from a laboratory-sized model have confirmed the conceptual model. The quantitative models for all parts of the process are based on the fundamental conservation equations of motion and thermodynamics, using a continuum approximation for each of the components.

The freezing of water to ice is a classic problem in applied mathematics, involving the solution of a diffusion equation with a moving boundary. However, when the water is salty, the transport of salt rejected by ice introduces some interesting twists to the tale. A number of analytic models for the freezing of water are briefly reviewed, ranging from the famous work by Neumann and Stefan in the 1800s, to the mushy zone models coming out of Cambridge and Oxford since the 1980s. The successes and limitations of these models, and remaining modelling issues, are considered in the case of freezing sea-water in the Arctic and Antarctic Oceans. A new, simple model which includes turbulent transport of heat and salt between ice and ocean is introduced and solved analytically, in two different cases—one where turbulence is given by a constant friction velocity, and the other where turbulence is buoyancy-driven and hence depends on ice thickness. Salt is found to play an important role, lowering interface temperatures, increasing oceanic heat flux, and slowing ice growth.

In this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi–Pasta–Ulam problem, a highly oscillatory Hamiltonian system. We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but at the wrong values. We use statistical properties such as time averages to further evaluate the performance of the trigonometric methods and compare the performance with that of the mid-point rule.

The current geothermal and volcanic activity in the North Island of New Zealand is explained as a consequence of Pacific and Australian plate interactions over the last 20 million years. The primary hypothesis is that the Kermadec subduction zone has for the last 20 million years or more been retreating in a south-easterly direction at about five centimetres per year. It is surmised that this motion and interaction with another subduction zone almost at right angles to it under the North Island resulted in plate tearing due to the incompatibility of the plate geometry where these subduction zones interacted. The nature and consequences of this plate tearing are partially revealed in published maps of the plate currently under the North Island. If the subducted parts of this plate, as shown in Eiby’s maps, [G. A. Eiby, “The New Zealand sub-crustal rift”, New Zeal. J. Geol. Geophy.7 (1964) 109–133] are straightened, then the plate edge lies on a curve giving a rough picture of their position before being torn and subducted by the Kermadec trench motion. This map of the tear suggests the shape of the edge of a missing plate segment torn from the plate, and implies a rotation of the upper North Island, clockwise approximately 20 degrees, about a point just south of the Thames estuary. A consequence of this plate tearing is that the solid retreating crustal wave generating magma pressure beneath the crest of the solid wave has the potential to inject significant basaltic magma into the crust through the tears. These intrusive magma fluxes have the ability to generate geothermal fields and rhyolitic lavas from crustal melts. This could explain the geothermal activity along the Coromandel peninsula five to seven million years ago, the ignimbrite outcrops about Lake Taupo and the current geothermal and volcanic activity stretching from Taupo to Rotorua.

In this paper, a weighted regularization method for the time-harmonic Maxwell equationswith perfect conducting or impedance boundary condition in composite materials is presented.The computational domain Ω is the unionof polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularitiesnear geometric singularities, which are the interior and exterior edges and corners. The variational formulation of theweighted regularized problem is given on the subspace of ${\cal H}$ ( ${\bf curl}$ ;Ω)whose fields $\textit{\textbf{ u}}$ satisfy $w^\alpha$ div ( $\varepsilon{\textit{\textbf{u}}}$ )∈L2 (Ω) and have vanishing tangential traceor tangential trace in L2 ( $\partial\Omega$ ). The weight function $w(\bf x)$ is equivalentto the distance of $\bf x$ to the geometric singularities and the minimal weight parameter αis given in terms of the singular exponents of a scalar transmission problem.A density result is proven that guarantees the approximability of the solution field by piecewise regular fields.Numerical results for the discretization of the source problemby means of Lagrange Finite Elements of type P 1 and P 2 are given onuniform and appropriately refined two-dimensional meshes.The performance of the method in the case of eigenvalue problems is addressed.

We present an approximate relation for the effective slip length for flows over mixed-slip surfaces with patterning at the nanoscale, whose minimum slip length is greater than the pattern length scale.

There are advantages in viewing orthogonal functions as functions generated by a random variable from a basis set of functions. Let Y be a random variable distributed uniformly on [0,1]. We give two ways of generating the Zernike radial polynomials with parameter l, {Zll+2n(x), n≥0}. The first is using the standard basis {xn,n≥0} and the random variable Y1/(l+1). The second is using the nonstandard basis {xl+2n,n≥0} and the random variable Y1/2. Zernike polynomials are important in the removal of lens aberrations, in characterizing video images with a small number of numbers, and in automatic aircraft identification.

Understanding ion transport in conjugated polymers is essential for developing mathematical models of applications of these materials. Previous experimental studies have suggested that cation transport in a conjugated polymer could be either diffusion or drift controlled, with debate over which dominates. In this paper we present a new model of cation transport that explains most of the features seen in a set of recent experiments. This model gives good agreement with measured concentration profiles, except when the profile has penetrated the polymer by more than 60%. The model shows that both diffusion and drift processes can be present. An application of a micro-actuator based on a conjugated polymer is presented to demonstrate that this technology could be used to develop a micro-pump.